심사위원장: 백형렬, 심사위원: 최서영, 박정환, 김상현(겸직교수), 이상진(건국대 수학과)

In recent years, local regularity theory for weak solutions to nonlocal equations with fractional orders has been studied extensively. In this talk, we discuss on local regularity for weak solutions to nonlocal equations with nonstandard growth and differentiability. In particular, we consider nonlocal equations of a variable exponent type, a double phase type and an Orlicz type.

The new infectious disease are emerging around the world. Coronavirus disease 2019 (COVID-19) caused by a novel coronavirus has emerged and has been rapidly spreading. The World Health Organization (WHO) declared the COVID-19 outbreak a global pandemic on March 11, 2020. Mathematical modelling plays a key role in interpreting the epidemiological data on the outbreak of infectious disease. Moreover, mathematical modeling can give us an early warning about the size of the outbreak. First, we construct a mathematical model to estimate the effective reproduction numbers, which assess the effect of control interventions. Second, we forecast the COVID-19 cases according to the different effect of control interventions. Finally, the most effective intervention can be suggested in terms of modeling approach. In this talk, I’d like to briefly introduce the main results of recent research on the mathematical modeling for various infectious diseases.

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ZOOM링크: https://kaist.zoom.us/j/84619675508

The talk with start with an introduction to Stark’s conjectures. We will then specialise to the situation of Brumer-Stark conjecture and its various refinements. I will then sketch a proof of the conjecture. This is a joint work with Samit Dasgupta.

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Please contact Wansu Kim at for Zoom meeting info or any inquiry.

Geometric and functional inequalities play a crucial role in several problems arising in analysis and geometry. Proving the validity of such inequalities, and understanding the structure of minimizers, is a classical and important question. In these lectures I will first give an overview of this beautiful topic and discuss some recent results.

We design and analyze V‐cycle multigrid methods for problems posed in H(div) and H(curl). Due to the fact that traditional smoothers do not work well for the vector field problems, special approaches for smoothers in the multigrid methods are essential. We introduce new smoothing techniques which involve non-overlapping domain decomposition preconditioners based on substructuring. We prove uniform convergence of the V‐cycle methods on bounded convex hexahedral domains. Numerical experiments that support the theory are also presented.

This series of lectures will focus on recent developments of the so-called a-contraction theory and its application to the study of discontinuous flow at high Reynolds numbers. We will first introduce the classical framework to study the stability of 1D shocks for compressible flows. Recent multi-D applications will be presented next, both in the context of compressible and incompressible flows.

Singular perturbations occur when a small coefficient affects the highest order derivatives in a system of partial differential equations. From the physical point of view, singular perturbations generate thin layers near the boundary of a domain, called boundary layers, where many important physical phenomena occur. In fluid mechanics, the Navier-Stokes equations, which describe the behavior of viscous flows, appear as a singular perturbation of the Euler equations for inviscid flows, where the small perturbation parameter is the viscosity. In general, verifying the convergence of the Navier-Stokes solutions to the Euler solution (known as the vanishing viscosity limit problem) remains an outstanding open question in mathematical physics. Up to now, it is not known if this vanishing viscosity limit holds true or not, even in 2D for which the existence, uniqueness, and regularity of solutions for all time are known for both the Navier-Stokes and Euler. In this talk, we discuss a recent result on the boundary layer analysis for the Navier-Stokes equations under a certain symmetry where the complete structure of boundary layers, vanishing viscosity limit, and vorticity accumulation on the boundary are investigated by using the method of correctors. We also discuss how to implement effective numerical schemes for slightly viscous fluid equations where the boundary layer correctors play essential roles. This is a joint work in part with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes, and with C.-Y. Jung and H. Lee.

I will give an introduction to the Monstrous moonshine conjectures of 70's-80's, which are on remarkable relations between Klein's j-invariant in number theory and the Monster sporadic simple group. I will only assume mild basic knowledge of complex analysis and group theory. I will start from a brief introduction to modular forms and Hauptmoduln, then connect it to finite simple groups. If I can manage the time, I will briefly explain a hint to a connection to the 3d gravity theory. https://kaist.zoom.us/j/84619675508

This series of lectures will focus on recent developments of the so-called a-contraction theory and its application to the study of discontinuous flow at high Reynolds numbers. We will first introduce the classical framework to study the stability of 1D shocks for compressible flows. Recent multi-D applications will be presented next, both in the context of compressible and incompressible flows.

In a recent joint work with Niudun Wang, we prove new results towards the Bhargava-Kane-Lenstra-Poonen-Rains conjectures on the first moment of Selmer groups over quadratic families of elliptic curves over global function fields. The key ingredients used in the proof are the Grothendieck-Lefschetz trace formula and zeroth homological stability of fiber bundles over configuration spaces. Both ideas form the backbone of a seminal work by Ellenberg, Venkatesh, and Westerland (2016), a rich incorporation of algebraic topological methods to arithmetic geometry. We shall give an overview of how these ideas are incorporated in analyzing the average size of Selmer groups, and examine how they can be implemented to approaching other arithmetic problems.

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Please contact Wansu Kim at for Zoom meeting info and any inquiry. For the list of Number Theory seminar talks, please visit the KAIST Number Theory seminar webpage. https://sites.google.com/site/wansukimmaths/kants-kaist-number-theory-seminar

In this talk, we present how to glue linear matrices in order to obtain a bigger linear matrix in a certain circumstance, and as a consequence, classify higher secant varieties of minimal degree. It is worth noting that by the del Pezzo-Bertini classification, a variety of minimal degree has determinantal presentation whenever its codimension is not small, and that higher secant varieties of minimal degree generalize varieties of minimal degree. This is a joint work with Prof. Sijong Kwak.

This series of lectures will focus on recent developments of the so-called a-contraction theory and its application to the study of discontinuous flow at high Reynolds numbers. We will first introduce the classical framework to study the stability of 1D shocks for compressible flows. Recent multi-D applications will be presented next, both in the context of compressible and incompressible flows.

Derived equivalence has been an interesting subject in relation to Fourier-Mukai transform, Hochschild homology, and algebraic K-theory, just to name a few. On the other hand, the attempt to classify schemes by their derived categories twisted by elements of Brauer groups is very restrictive as we have a positive answer only for affines. I'll talk about how we can extend this result to a broader class of algebro-geometric objects in the setting of derived/spectral algebraic geometry at the expense of a stronger notion of twisted equivalences than that of ordinary twisted derived equivalences. I'll convince you that the new notion is not only reasonable, but also indispensable from this point of view. The second talk will be dedicated to studying twisted derived equivalences in the derived/spectral setting. As a consequence, a derived/spectral analogue of Rickard's theorem, which shows that derived equivalent associative rings have isomorphic centers, will be discussed. I'll try to avoid technicalities related to using the language of derived/spectral algebraic geometry.

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Zoom ID: 352 730 6970, Password: 9999. You will be authorized individually by the host of the meeting.

In this talk I will consider the spectral gap for the linearized Boltzmann or Landau equation with soft potentials. It is known that the corresponding collision operators admit only the degenerated spectral gap. We rather prove the formation of spectral gap in the spatially inhomogeneous setting where the space domain is bounded with an inflow boundary condition. The key strategy is to introduce a new Hilbert space with an exponential weight function that involves the inner product of space and velocity variables and also has the strictly positive upper and lower bounds. The action of the transport operator on such space-velocity dependent weight function induces an extra non-degenerate relaxation dissipation in large velocity that can be employed to compensate the degenerate spectral gap and hence give the exponential decay for solutions in contrast with the sub-exponential decay in either the spatially homogeneous case or the case of torus domain. The result reveals a new insight of hypocoercivity for kinetic equations with soft potentials in the specified situation.

In this talk we consider the Waring rank of monomials over the rational numbers. We give a new upper bound for it by establishing a way in which one can take a structured apolar set for any given monomial. This bound coincides with all the known cases for the real rank of monomials, and is sharper than any other known bounds for the real Waring rank. Since all of the constructions are still valid over the rational numbers, this provides a new result for the rational Waring rank of any monomial as well. We also apply the methods developed in the paper to the problem of finding an explicit rational Waring decomposition of any homogeneous polynomial over rational numbers, which is important in many applications, especially to the integration of a polynomial over a simplex. We will present examples and computational implementation for potential use.

We formulate, and provide strong evidence for, a natural generalization of a conjecture of Robert Coleman concerning higher rank Euler systems for the multiplicative group over arbitrary number fields. This is a joint work with Burns, Daoud, and Sano.

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Please contact Wansu Kim at for Zoom meeting info and any inquiry. For the list of Number Theory seminar talks, please visit the KAIST Number Theory seminar webpage. https://sites.google.com/site/wansukimmaths/kants-kaist-number-theory-seminar

Let  C⊂P^r be a nondegenerate projective integral curve of degree d and arithmetic genus g. A celebrated theorem of Castelnuovo gives an explicit upper bound pi_0(d,r) on g in terms of d and n. Moreover, if d ≥ 2r+1 then g=pi_0 (d,r) if and only if C is ACM and it lies on a surface of minimal degree. In 1980, Joe Harris and David Eisenbud proved that (i) C lies on a surface of minimal degree if g> pi_1 (d,r), and (ii) if g=pi_1(d,r) and C does not lie on a surface of minimal degree, then there exists a del Pezzo surface which contains C. Along this line, we will show that there exists an integer pi_1(d,r)^' < pi_1(d,r) such that  C lies on a del Pezzo surface if g> pi_1(d,r)^' This is a joint work with Wanseok Lee

This series of lectures will focus on recent developments of the so-called a-contraction theory and its application to the study of discontinuous flow at high Reynolds numbers. We will first introduce the classical framework to study the stability of 1D shocks for compressible flows. Recent multi-D applications will be presented next, both in the context of compressible and incompressible flows.

KAIX Distinguished lectures in Mathematics Speaker : Wen-Ching Winnie Li (Distinguished Professor of Mathematics, Penn. State Univ.) 2021.11.09 (Tue) - Korean time 09:30-10:30 Colloquium talk Primes in Number Theory and Combinatorics 10:30-10:50 Q&A 11:00-12:00(noon) Seminar Talk Pair arithmetical equivalence for quadratic fields ZOOM ID : 518 127 6292 (No password required) Abstract: 1. colloquium talk Title: Primes in number theory and combinatorics Abstract: Prime numbers are a central topic in number theory. They have inspired the study of many subjects in mathematics. Regarding prime numbers as the building blocks of the multiplicative structure of positive integers, in this survey talk we shall interpret "primes" as the basic elements in a structure of interest arising from combinatorics and number theory, and explore their distributions of various kinds. More precisely, we shall examine primes in compact Riemann surfaces, graphs, and 2-dimensional simplicial complexes, respectively. These results are products of rich interplay between number theory and combinatorics. 2. number theory seminar talk Title: Pair arithmetical equivalence for quadratic fields Abstract: Given two nonisomorphic number fields K and M, and two finite order Hecke characters $\chi$ of K and $\eta$ of M respectively, we say that the pairs $(\chi, K)$ and $(\eta, M)$ are arithmetically equivalent if the associated L-functions coincide: $L(s, \chi, K) = L(s, \eta, M)$. When the characters are trivial, this reduces to the question of fields with the same Dedekind zeta function, investigated by Gassmann in 1926, who found such fields of degree 180, and by Perlis in 1977 and others, who showed that there are no arithmetically equivalent fields of degree less than 7. In this talk we discuss arithmetically equivalent pairs where the fields are quadratic. They give rise to dihedral automorphic forms induced from characters of different quadratic fields. We characterize when a given pair is arithmetically equivalent to another pair, explicitly construct such pairs for infinitely many quadratic extensions with odd class number, and classify such characters of order 2. This is a joint work with Zeev Rudnick.

In my next talk, I will define canonical dimension of varieties (which, roughly speaking, measures how hard it is to get a rational point in a given variety) and canonical dimension of algebraic groups (which, roughly speaking, measures how complicated the torrsors of an algebraic group can be). Then I will state several previously known facts from intersection theory and from theory of canonical dimension, and I will prove that if we know that a certain product of Schubert divisors is mutiplicity-free (which was defined in my first talk), then this fact implies an upper estimate on the canonical dimension of the group and its torsors. As a result, we will get some explicit numerical estimates on canonical dimension of simply connected simple split algebraic groups groups with simply-laced Dynkin diagrams.

Derived equivalence has been an interesting subject in relation to Fourier-Mukai transform, Hochschild homology, and algebraic K-theory, just to name a few. On the other hand, the attempt to classify schemes by their derived categories twisted by elements of Brauer groups is very restrictive as we have a positive answer only for affines. I'll talk about how we can extend this result to a broader class of algebro-geometric objects in the setting of derived/spectral algebraic geometry at the expense of a stronger notion of twisted equivalences than that of ordinary twisted derived equivalences. I'll convince you that the new notion is not only reasonable, but also indispensable from this point of view. The first talk will be mainly devoted to giving brief expository accounts of some background materials needed to understand the notion of twisted derived equivalence in the setting of derived/spectral algebraic geometry; in particular, some familiarity with ordinary algebraic geometry will be enough for the talk.

We will survey recent development in subadditive thermodynamic formalism for matrix cocycles. In particular, in the setting of locally constant cocycles as well as fiber-bunched cocycles, we will discuss sufficient conditions for the norm potentials of such cocycles to have unique equilibrium states. If time permitting, we will also discuss ergodic properties of such equilibrium states as well as some applications.

We introduce a distribution-theoretic conjecture of Roert Coleman of the 1980's and prove the conjecture in a recent joint work with Burns and Daoud. This accordingly gives an explicit description of the complete set of Euler systems for the multiplicative group over Q together with a connection to other conjectures in number theory.

In this talk, we present the hydrodynamic limits of the Schrodinger equation, affected by different gauge fields. Precisely, we first present the hydrodynamic limit of the Schrodinger equation with the Chern-Simons gauge fields (Chern-Simons-Schrodinger equation), toward to the Euler-Chern-Simons equation on the two-dimensional state space. Then, we consider the hydrodynamic limit of the Schrodinger equation with the Maxwell gauge fields (Maxwell-Schrodinger equation), toward to the Euler-Maxwell equation on the three-dimensional state space. Both estimate use the estimate on the modulated energy functionals.

심사위원장 최건호, 심사위원 곽도영, 권순식, 김동석(경영공학부), 김완수

First, I will say a few words about Galois descent in the particular case of a projective variety embedded into a projective space. Then I will recall the definintion of a torsor and will explain how to construct the quotient of a torsor of a simple simply connected split algebraic group modulo a Borel subgroup. Finally, I will prove that the Picard group of such a quotient does not change for one particular finite Galois extension of the base field, and then, if there is enough time, for any extension of the base field.

KAIX 석학 강연

KAIX 석학강연

The purpose of this talk is to mathematically investigate the formation of a plasma sheath, and to analyze the Bohm criterions which are required for the formation. Bohm derived originally the (hydrodynamic) Bohm criterion from the Euler–Poisson system. Boyd and Thompson proposed the (kinetic) Bohm criterion from kinetic point of view, and then Riemann derived it from the Vlasov–Poisson system. We study the solvability of boundary value problems of the Vlasov–Poisson system. On the process, we see that the kinetic Bohm criterion is a necessary condition for the solvability. The argument gives a simpler derivation of the criterion. Furthermore, the hydrodynamic criterion can be derived from the kinetic criterion. It is of great interest to find the relation between the solutions of the Vlasov–Poisson and Euler–Poisson systems. To clarify the relation, we also investigate the hydrodynamic limit of solutions of the Vlasov–Poisson system.

The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we give a new lower bound on the smooth nonorientable four-ball genus $\gamma_4$ of any knot. This bound is sharp for several families of torus knots, including $T_{4n,(2n\pm 1)^2}$ for even $n\ge2$, a family Longo showed were counterexamples to Batson's conjecture. We also prove that, whenever $p$ is an even positive integer and $\frac{p}{2}$ is not a perfect square, the torus knot $T_{p,q}$ does not bound a locally flat M{\" o}bius band for almost all integers $q$ relatively prime to $p$.

Anyons are quasiparticles in two dimensions. They do not belong to the two classes of elementary particles, bosons and fermions. Instead, they obey Abelian or non-Abelian fractional statistics. Their quantum mechanical states are determined by fusion or braiding, to which braid groups and conformal field theories are naturally applied. Some of non-Abelian anyons are central in realization of topological qubits and topological quantum computing. I will introduce the basic properties of anyons and their recent experimental signatures observed in systems of topological order such as fractional quantum Hall systems and topological superconductors.

The cohomology of local Shimura varieties, and of more general spaces of local shtukas, is of fundamental interest in the Langlands program. On the one hand, it is supposed to realize instances of the local Langlands correspondence. On the other hand, there is a tight relationship with the cohomology of global Shimura varieties. In recent joint work with Kaletha and Weinstein, we proved the first general results towards the Kottwitz conjecture, which predicts how supercuspidal L-packets contribute to the cohomology of local shtuka spaces. I will review this whole story, and give some overview of the ideas which enter into our proof. The key idea in our argument - namely, that the Kottwitz conjecture should follow from some form of the Lefschetz-Verdier fixed point formula - was already formulated by Michael Harris in the '90s. However, executing this idea brings substantial technical challenges. I will try to emphasize the new ingredients which allow us to implement this idea in full generality.

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Please contact Wansu Kim at for Zoom meeting info and any inquiry. For the list of Number Theory seminar talks, please visit the KAIST Number Theory seminar webpage. https://sites.google.com/site/wansukimmaths/kants-kaist-number-theory-seminar

In the first part, I introduce a novel variational model for the joint enhancement and restoration of dark images corrupted by blurring and/or noise. The model decomposes a given dark image into reflectance and illumination images that are recovered from blurring and/or noise. In addition, our approach utilizes non-convex total variation regularization on all variables. This allows us to adequately denoise homogeneous regions while preserving the details and edges in both reflectance and illumination images, which leads to clean and sharp final enhanced images. Experimental results demonstrate the effectiveness of the proposed model when compared to other state-of-the-art methods in terms of both visual aspect and image quality measures. In the second part, I propose a novel variational model for the restoration of a single color image degenerated by haze. The model extends the total variation based model, by inserting an inter-channel correlation term. This additional term permits both color and gray-valued transmission maps, which enable broader applications of the proposed model. Numerical experiments validate the outstanding performance of the proposed model compared to the state-of-the-art methods.

The next few talks will be more like learning than research: I will explain some preparation material, which is considered "well-known" by the experts, but which I didn't find a reference for in the form I need. My next goal is to explain the proof that the Picard group of the so-called quotient of a torsor of a simply connected simple split algebraic group modulo a Borel subgroup does not change under field extension. In the first talk I will explain the basic machinery to prove this fact, namely Galois descent theory. Given a variety X over a non-algebraically closed field F with no or "not enough" rational points, Galois descent theory allows one to work with an extension K of F and with X_K and study the properties of the original X. If there is enough time, I will also define torsors and show how to construct them using Galois descent.

In this talk, we are going to discuss boundary regularities of various degenerate local equation and nonlocal equations. Diffusion rates deform undefined geometry related to diffusion and the corresponding distance function makes important role in the theory of regularity. And then we will also discuss the possible applications.

The temporal credit assignment, the problem of determining which actions in the past are responsible for the current outcome (long-term cause and effect), is difficult to solve because one needs to backpropagate the error signal through space and time. Despite its computational challenges, humans are very good at solving this problem. Our lab uses reinforcement learning theory and algorithms to explore the nature of computations underlying the brain’s ability to solve the temporal credit assignment. I will outline two-fold approaches to this issue: (1) training a computational model from human behavioral data without underfitting and overfitting (Brain → AI) and (2) using the trained model to manipulate the way the human brain solves the temporal credit assignment problem (AI → brain).

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Education/employments PhD, KAIST (2009)Postdoc, MIT (2010-2011), Caltech (2011-2015)Faculty, KAIST (2015-now) Honors/awards IBM Academic Research Award (2021)Google Faculty Research Award (2017)Della-Martin Fellowship (2014) KAIST Breakthroughs (2020)KAIST Songam Distinguished Research Award (2019)KAIST Top 10 Technologies (2019)KAIST Institute Faculty Award (2019) KIIS Young Investigator Award (2016)ICROS Young Investigator Award (2016)

Given a sequence of random i.i.d. 2 by 2 complex matrices, it is a classical problem to study the statistical properties of their product. This theory dates back to fundamental works of Furstenberg, Kesten, etc. and is still an active research topic. In this talk, I intend to show how methods from complex analysis and analogies with holomorphic dynamics offer a new point of view to this problem. This is used to obtain several new limit theorems for these random processes, often in their optimal version. This is based on joint works with T.-C. Dinh and H. Wu.

We continue our discussion on the result of Marden, Thurston and Bonahon which states that in hyperbolic 3-manifolds, every immersed surface of which the fundamental group is invectively embedded in the 3-manifold group is quasi-fuchsian or doubly degenerated. Surface subgroups of 3-manifold groups play an important rule in 3-manifold theory. For instance, some collection of immersed surfaces give rise to a CAT(0) cube complex. Especially, in the usual construction of the CAT(0) cube complex, each immersed surface composing the collection is quasi-fuchsian. In this talk, I introduce the work by Cooper, Long and Reid. In hyperbolic mapping tori, the work gives a criterion to determine whether the given immersed surface is quasi-fuchsian or not. The criterion is given in terms of laminations induced in immersed surfaces.

For a quadratic projective variety X ⊂P_r , the locus of quadratic polynomials of rank 3 in the homogeneous ideal I(X) defines a projective algebraic set, say PHI_3(X), in P(I, (X)_2). So, it provides several projective invariants of X. In this talk, I will speak about the structure of Phi_2(C) when C ⊂P_n is a rational normal curve. This is based on the joint work with Saerom Shim.

In his famous 1900 presentation, Hilbert proposed so-called the Hilbert’s 6thproblem, namely “Mathematical Treatment of the Axioms of Physics”. He mentioned that “Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua.” In this lecture, we present some recent development of the Hilbert’s 6th problem in the Boltzmann theory when the various fluid models have natural “singularities” such as unbounded vorticity and formation of boundary layers.

이번 발표를 통해 수리과학 모델을 이용한 감염병 확산 예측 방법 그리고 방역 정책의 감염 확산 억제 효과 분석에 대하여 소개하겠습니다. 감염 확산 모형의 가장 기본이면서 널리 쓰이고 있는 compartment model, 그리고 지역 단위 인구 이동 자료를 반영한 metapopulation model에 대하여 논의하고, 시시각각 변화하는 감염 확산 상황을 표현하기 적합한 data assimilation method을 살펴보겠습니다. 방역 정책 효과 분석을 위한 수리과학 모델로서 microsimulation model을 소개하겠습니다. Microsimulation model은 정부의 정책 변화가 사회, 경제적으로 미치는 영향을 분석하고자 제안된 시뮬레이션 도구로 거시적 수준의 경제, 사회, 인구 변화를 각 개인과 가구 단위의 미시적 사건들로부터 기술합니다. Microsimulation model을 이용하면 가구, 직장/학교, 종교 및 친목 모임의 밀접 접촉을 통한 호흡기 감염병 확산을 시뮬레이션할 수 있습니다. 그리고 휴교령, 직장 재택 근무, 종교 시설 폐쇄 등의 비약물적 조치가 감염병 확산 방지에 어떤 효과를 지니는지 분석할 수 있다는 장점이 있습니다.

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https://us02web.zoom.us/j/82312487069?pwd=RUJFUmVaZnBYdzJNOUZ5TTRIbzJXZz09

The cohomology of Shimura varieties have rich structures and have been studied for many years. Some new vanishing theorems were proved in the last few years and especially the one by Caraiani-Scholze is crucial in arithmetic applications. I will survey these results, and discuss further development.

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(Please contact Wansu Kim at for Zoom meeting info and any inquiry.)

In my first talk I am going to speak about Schubert calculus. Let G/B be a flag variety, where G is a linear simple algebraic group, and B is a Borel subgroup. Schubert calculus studies (in classical terms) multiplication in the cohomology ring of a flag variety over the complex numbers, or (in more algebraic terms) the Chow ring of the flag variety. This ring is generated as a group by the classes of so-called Schubert varieties (or their Poincare duals, if we speak about the classical cohomology ring), i. e. of the varieties of the form BwB/B, where w is an element of the Weyl group. As a ring, it is almost generated by the classes of Schubert varieties of codimension 1, called Schubert divisors. More precisely, the subring generated by Schubert divisors is a subgroup of finite index. These two facts lead to the following general question: how to decompose a product of Schubert divisors into a linear combination of Schubert varieties. In my talk, I am going to address (and answer if I have time) two more particular versions of this question: If G is of type A, D, or E, when does a coefficient in such a linear combination equal 0? When does it equal 1?

This is joint work with Kenjiro Ishizuka (Kyoto). We study global behavior of solutions to the nonlinear Klein-Gordon equation with a damping and a focusing nonlinearity on the Euclidean space. Recently, Cote, Martel and Yuan proved the soliton resolution conjecture completely in the one-dimensional case: every global solution in the energy space is asymptotic to a superposition of solitons getting away from each other as time tends to infinity. The next question is to see which initial data evolve into each of the asymptotic forms. The asymptotic decomposition is very sensitive to initial perturbation because all the solitons are unstable. We consider the simplest non-trivial setting in general space dimensions: the global behavior of solutions starting near a superposition of two ground states. Cote, Martel, Yuan and Zhao proved that the solutions asymptotic to 2-solitons form a codimension-2 manifold in the energy space. Our question is what happens for the other initial data in the neighborhood. As an answer, we give a complete classification of those solutions into 5 types of global behavior. Two of them are asymptotic to the positive ground state and the negative one respectively. They form two codimension-1 manifolds that are joined at their boundary by the Cote-Martel-Yuan-Zhao manifold of 2-solitons. The connected union of those three manifolds separates the remainder of the neighborhood into the open set of global decaying solutions and that of blow-up. The main difficulty to prove it is in controlling the direction of instability in two dimensions attached to the two soliton components, because the soliton interactions are not integrable in time, breaking the simple superposition of the linearized approximation around each soliton. It is resolved by showing that the non-integrable interactions do not essentially affect the direction of instability, using the reflection symmetry of the equation and the 2-solitons. I will also explain the difficulty for the 3-solitons due to a more dramatic phenomenon, which may be called soliton merger.